Initial volume of the cube,

V = Length x width x height = L3

Now we will secure here the final dimensions of the cube in order to secure the final volume of the cube and finally we will determine the bulk modulus of elasticity.

Let us consider first one side of cube i.e. AB. As we have already discussed that three mutually perpendicular tensile stresses of similar intensity are acting over the cube. Let us determine here the effect of tensile stress over the dimensions of the cube.

As we have already seen that, Ԑ = dL/L

Strain = dL/L

dL= L x Stress x α = L x σ x α

dL= L. σ. α

Now we will have to think slightly here to discuss the effect on length of the cube under three mutually perpendicular tensile stresses of similar intensity. When direct tensile stress will be subjected over the face AEHD and BFGC, there will be increase in length due to longitudinal strain developed due to direct tensile stress acting over the face AEHD and BFGC.

Simultaneously, we must have to note it here that tensile stress acting over the face AEFB and DHGC will develop the lateral strain in side AB.

Similarly, tensile stress acting over the face ABCD and EFGH will also develop the strain in side AB

Final length of the cube, = L + L. σ. α – L. σ. β - L. σ. β

Final side length of the cube, = L [1 + σ. (α – 2β)]

Final Volume of the cube

Vf = L3 x [1 + σ. (α – 2β)] 3

Now we will ignore the product of small quantities in order to easy understanding

Vf = L3 x [1 + σ. (α – 2β)] 3

Vf = L3 + 3 σ. L3 (α – 2β)

Change in volume of the cube, when three mutually perpendicular tensile stresses of similar intensity are acting over the cube.

ΔV = L3 + 3 σ. L3 (α – 2β) - L3

ΔV = 3 σ. L3 (α – 2β)

Let us see here volumetric strain

Volumetric strain in the specified cube here will be determined as displayed here

Volumetric strain = ΔV/V

ԐV = 3 σ (α – 2β)

Now, we will find here Bulk modulus of elasticity (K)

Bulk modulus of elasticity will be defined as the ratio of volumetric stress or hydro static stress to volumetric strain and therefore we will write here as mentioned here

K = σ / [3 σ (α – 2β)]

K =1/ [3 (α – 2β)]

3 K (α – 2β) = 1

3K (1-2 β/α) = 1/ α

As we have already seen above that

Young’s modulus of elasticity, E = 1/ α

Poisson ratio, ν = (β/α)

After replacing the value of 1/ α and (β/α) in above concluded equation, we will have the desired result which will show the relationship between young’s modulus of elasticity (E) and bulk modulus of elasticity (K)

3K (1-2 ν) = E

E = 3K (1-2 ν)

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